Mahler’s “Der Kindertotenlieder” if classical counts.
Sorry to post again, but I found a link for mine.
Goodbye — Elton live in 1971
Youtube can be an absolute goldmine of nostalgia. 
Really? I always thought it was kind of uplifting. I do feel a little sad whenever I hear it, but I think that’s just because I associate it with someone with whom I had an odd sort of relationship. Either way, though, I think it’s a beautiful song, capable of evoking emotions of any kind.
My nomination is “Hey You” by Pink Floyd. For me, I knew, it had to be Pink Floyd. The question was just which of their songs was depressing enough.
James Taylor’s Long Ago and Far Away.
Thanks for sharing, that was a fine tune, indeed.
My contestant is Bob Dylan’s Not Dark Yet .
Down in a Hole by Alice in Chains is pretty damned depressing.
Hank Williams “I’m So Lonesome I Could Cry”
The Yardbirds’ “Still I’m Sad” is right up there in the depression Hall of Fame.
It’s sung in the style of a Gregorian Chant:
See the stars come falling down from the sky.
Gently passing, they kiss your tears when you cry.
See the wind come softly blow the hair from your face.
See the rain hide away in disgrace.
Still I’m sad. How I’m sad.
There is a second verse but I won’t type it because:
- SDMB policy forbids the printing of a song’s entire lyrics (copyright issues).
- I don’t feel like supplying the readers of this thread with Prozac™.
Incidentally, I think this is post # 5,000 for me. 
You’re right, overall the song does have an uplifting message, but still it makes me sad. The other songs on the album are actually a lot more grim.
Gorecki - 3rd . I’m pretty sure this is Dawn Upshaw singing and although the video is bits from from Leon, it’s lovely.
This is the song I came in here to mention. The first time I heard it I had to choke back tears.
A whole slew of songs by The Smiths would have to make the list, notably Asleep and Please, Please, Please Let Me Get What I Want
Nodding in acceptance to many songs here, my own addition is Sufjan Stevens’ Casimir Pulaski Day.
Neil Young: The Needle and the Damage Done
This one wrecks me every time.
Carpenters, Goodbye to Love. 
I’m gonna have to go with Dan Fogelberg’s Same Old Lang Syne. Man, that song just depresses me every time I hear it.
Roads to Moscow by Al Stewart
How about a song that has been the subject of many, many (many, many) threads on this board: Ode to Billy Joe by Bobbie Gentry?